Short answer: yes
Long answer: In some, but not all cases. It is up to the Senate.
The Constitution mandates removal from office upon conviction, but specifies that an additional penalty, being barred from any office of honor in the USA, is possible but not mandatory.
In US history, eight non-presidential office holders have been convicted in a Senate impeachment trial. In only three of those cases was the convicted party barred from holding future offices. In the other five cases, the Senate voted for removal only. In one of those instances, the impeached official went on to a long post-impeachment career in politics. Federal district judge Alcee Hastings of Florida was removed from office in 1989 for perjury and conspiring to solicit a bribe. Since 1993, he has been representing a Florida district in the U.S. House of Representatives.